It seems at first sight that the aberration of light and the related optical and electrical phenomena will provide us a means of determining the absolute motion of the Earth, or rather its motion, not in relation to the other stars, but in relation to the ether. Fresnel had already tried it, but he recognized soon that the motion of the earth does not alter the laws of refraction and reflection. Similar experiments, like that of a telescope filled with water and all those which take into consideration only terms of first order in respect to aberration, give no other but negative results; soon an explanation was discovered; but Michelson, having imagined an experiment where the terms depending on the square of the aberration became sensitive, failed as well.

It seems that this impossibility of demonstrating an experimental evidence for absolute motion of the Earth is a general law of nature; we are naturally led to admit this law, which we will call the Postulate of Relativity and admit it without restriction. This postulate, which is up to now in accord with experiments, may be either confirmed or disproved later by more precise experiments, it is in any case interesting to see which consequences follow from it.

An explanation was proposed by Lorentz and FitzGerald, who introduced the hypothesis of a contraction undergone by all bodies into the direction of the motion of earth and proportional to the square of aberration; this contraction, which we will call Lorentz contraction, would give an account of the experiment of Michelson and all those which were carried out up to now. The hypothesis would become insufficient, however, if one were to assume the postulate of relativity in all its generality.

The importance of the question determined me to take it up again; the results which I obtained are in agreement with those of Lorentz on all important points; I was only led to modify and supplement them in some points of detail; one will further see the differences which are of secondary importance.

The idea of Lorentz can be summarized as follows: if we can bring the whole system to a common translation, without modification of any of the apparent phenomena, it is because the equations of the electromagnetic medium are not altered by certain transformations, which we will call Lorentz transformation; two systems, one motionless, the other in translation, thus become exact images of one another.

Langevin[1] had sought to modify the idea of Lorentz; for both authors the moving electron takes the shape of a flattened ellipsoid, but for Lorentz two of the axes of the ellipsoid remain constant, while for Langevin on the contrary it is the volume of the ellipsoid which remains constant. Besides, both scientists showed hat these two hypothesis are in agreement with the experiments of Kaufmann, as well as the original hypothesis of Abraham (undeformable spherical electron).

The advantage of the theory of Langevin is that it uses only electromagnetic forces and binding forces; but it is incompatible with the postulate of relativity; this is what Lorentz had shown, this is what I find again in another way by relying upon the principles of group theory.

It is thus necessary to return from here to the theory of Lorentz; but if one wants to preserve it and avoid intolerable contradictions, it is necessary to suppose a special force which explains at the same time the contraction and the constancy of two of the axes. I sought to determine this force, I found that it can be compared to a constant external pressure, acting on the deformable and compressible electron, and whose work is proportional to the variations of the volume of the electron.

So if the inertia of matter is exclusively of electromagnetic origin, as it is generally admitted since the experiment of Kaufmann, and except that constant pressure from which I come to speak, all forces are of electromagnetic origin, the postulate of relativity can be established in any rigour. It is what I show by a very simple calculation founded on the principle of least action.

But this is not all. Lorentz, in the quoted work, considered it to be necessary to supplement his hypothesis so that the postulate remains when there are other forces as the electromagnetic forces. According to him, all the forces, whatever is their origin, are affected by the Lorentz transformation (and consequently by a translation) in the same way as the electromagnetic forces.

It was important to examine this assumption more closely and in particular to seek which modifications it would oblige us to bring to the laws of gravitation.

It is found at first sight, that we are forced to suppose that the propagation of gravitation is not instantaneous, but happens with the speed of light. One could believe that this is a sufficient reason to reject the hypothesis, as Laplace has shown that this cannot be so. But actually, this propagation effect is mainly compensated by a different cause, so that there is no more contradiction between the proposed law and the astronomical observations.

Is it possible to find a law, which satisfies the condition imposed by Lorentz, and which at the same time is reduced to the law of Newton when the speeds of the stars are rather small, so that one can neglect their squares (as well as the product of acceleration and distance) in respect to the square speed of light?

To this question, as it further will be seen, one must answer in the affirmative.

Is the law thus amended compatible with the astronomical observations?

At first sight it seems that it is the case, but this question can be decided only by a thorough discussion.

But even accepting that the discussion turns to the advantage of a new hypothesis, what should we conclude? If the propagation of attraction happens with the speed of light, it cannot be by a fortuitous coincidence, it must be due to a function of the ether; and then it will be necessary to seek to penetrate the nature of this function, and to relate it to the other functions of the fluid.

We cannot be satisfied with simply juxtaposed formulas which would agree only by a lucky stroke; it is necessary that these formulas are so to speak able to be penetrated mutually. Our mind will not be satisfied before it believes to see the reason of this agreement, at the point where it has the illusion that it could have predicted it.

But the question can still be seen form another point of view, which could be better understood by analogy. Let us suppose an astronomer before Copernicus who reflects on the system of Ptolemy; he will notice that for all planets one of the two circles, epicycle or deferent, is traversed in the same time. This cannot be by chance, there is thus between all planets a mysterious binding.

But Copernicus, by simply changing the axes of coordinates regarded as fixed, destroyed this appearance; each planet does not describe any more than only one circle and the durations of the revolutions become independent (until Kepler restores between them the binding which was believed to be destroyed).

Here it is possible that there is something analogue; if we admit the postulate of relativity, we would find in the law of gravitation and the electromagnetic laws a common number which would be the speed of light; and we would still find it in all the other forces of any origin, which could be explained only in two manners:

Either there would be nothing in the world which is not of electromagnetic origin.

Or this part which would be, so to speak, common to all the physical phenomena, would be only apparent, something which would be due to our methods of measurement. How do we perform our measurements? By transportation, one on the other, of objects regarded as invariable solids, one will answer immediately; but this is not true any more in the current theory, if the Lorentz contraction is admitted. In this theory, two equal lengths are, by definition, two lengths for which light takes the same time to traverse.

Perhaps it would be enough to give up this definition, so that the theory of Lorentz is as completely rejected as it was the system of Ptolemy by the intervention of Copernicus. If that happens one day, it will not prove that the effort made by Lorentz was useless; because Ptolemy, no matter what we think about him, was not useless for Copernicus.

Also I did not hesitate to publish these few partial results, although in this moment even the whole theory seems to be endangered by the discovery of magnetocathodic rays.

Lorentz had adopted a particular system of units, so as to eliminate the factors 4π in the formulas. I'll do the same, plus I choose the units of length and time so that the speed of light is equal to 1. Under these conditions the fundamental formulas become (by calling f, g, h the electric displacement, α, β, γ the magnetic force, F, G and H the vector potential, φ the scalar potential, ρ the electric density, ξ, η, ζ the electron velocity, u, v, w the current):

These equations are capable of a remarkable transformation discovered by Lorentz and which owes its interest from the fact, that it explains why no experience is suited to show us the absolute motion of the universe. Let:

are related to the functions (5) by the same linear relations as the old variables to the new variables. Then, if we denote by D' the functional determinant of the functions (5bis) in relation to the new variables, we have:

Our transformation does not alter the equations (I). Indeed, the continuity condition, and the equations (6) and (8), already provided us with some of the equations (I) (except the accentuation of letters).

If we represent the components of the force X1, Y1, Z1, not per unit volume, but per unit of electric charge of the electron, and X'1, Y'1, Z'1 are the same quantities after the transformation, we would have:

Before going further, it is important to investigate the cause of this significant discrepancy. It is obvious that the formulas for ξ', η', ζ' are not the same, while the formulas for the electric and magnetic fields are the same.

If the inertia of electrons is exclusively of electromagnetic origin, if in addition they are subject only to forces of electromagnetic origin, the equilibrium condition requires that we have inside the electrons:

X=Y=Z=0.{\displaystyle X=Y=Z=0.\,}

But in virtue of equations (11) those relations are equivalent to

X′=Y′=Z′=0.{\displaystyle X'=Y'=Z'=0.\,}

The equilibrium conditions of the electrons are not altered by the transformation.

Unfortunately, a hypothesis as simple as that is unacceptable. If, indeed, we assume ξ=η=ζ=0{\displaystyle \xi =\eta =\zeta =0}, the conditions X=Y=Z=0{\displaystyle X=Y=Z=0} entrain f=g=h=0{\displaystyle f=g=h=0}, and consequently ∑dfdx=0{\displaystyle \sum {\tfrac {df}{dx}}=0}, i.e. ρ = 0. We arrive at similar results in the most general case. We must therefore admit that there are, in addition to electromagnetic forces, either other forces or bindings. It is necessary to search for conditions which must satisfy these forces or bindings, so that the equilibrium of the electron is not disturbed by the transformation. This will be the subject of a later paragraph.

We know how Lorentz deduced his equations from the principle of least action. I will return to this question, even though I have nothing substantial to add to the analysis of Lorentz, because I prefer to present it in a slightly different form which will be useful for my purpose. I will pose:

Our integrations are extended to infinity, it must be x=±∞{\displaystyle x=\pm \infty } in the first integral on the right-hand side; so, since we always assume that all our functions vanish at infinity, this integral will be zero and it follows

The principles of variation calculus tells us that we must do the calculation as if ψ is an arbitrary function, as if δJ is represented by (6), and as if the changes were no longer subject to the condition (5).

It remains to determine the forces acting on the electrons. To do this we must suppose that a supplementary force -Xdτ, -Ydτ, -Zdτ applies to each element of an electron, and write that this force is in equilibrium with the forces of electromagnetic origin. Let U, V, W be components of the displacement of the element dτ of the electron, where the displacement is counted from an arbitrary initial position. Let δU, δV, δW be the variations of this displacement; the virtual work corresponding to the supplementary force is:

−∫∑XδUdτ,{\displaystyle -\int \sum X\delta U\ d\tau ,}

so that the equilibrium condition about which we have spoken can be written:

It is indeed convenient to switch from the notation of variation calculus to that of ordinary calculus, or vice versa.

Our functions should be regarded: 1° as dependent on five variables x, y, z, t, ε, so that we can remain at the same place when ε and t vary alone: we then indicate their derivatives by the ordinary d; 2° as dependent on five variables x0, y0, z0, t, ε so that we may always follow a single electron when t and ε vary alone, then we denote their derivatives by ∂. We will have then:

Note the difference between the definition of δU=dUdϵδϵ{\displaystyle \delta U={\tfrac {dU}{d\epsilon }}\delta \epsilon } and that of δρ=dρdϵδϵ{\displaystyle \delta \rho ={\tfrac {d\rho }{d\epsilon }}\delta \epsilon }, we note that it is this definition of δU that suits to formula (10).

This equation will allow us to transform the first term of (9); we find in fact:

However, to justify this equality, the integration limits have to be the same; so far we have assumed that t varies from t0 to t1, and x, y, z from ∞ to + ∞. On this account the integration limits would be affected by the Lorentz transformation, but nothing prevents us from assuming t0 =- ∞, t1 = + ∞; with those conditions the limits are the same for J and J'.

We then compare the following two equations analogues to equation (10) of § 2:

These are the equations (11) of § 1. The principle of least action leads us to the same result as the analysis of § 1.

If we turn to formulas (1), we see that Σf² - Σα² is not affected by the Lorentz transformation, except one constant factor; it is not the case with expression Σf² + Σα² which represents the energy. If we confine ourselves to the case where ε is sufficiently small, so that the square can be neglected so that k = 1, and if we also assume l = 1, we find:

but it is easy to see that this transformation is equivalent to a coordinate change, the axes are rotating a very small angle around the z-axis. We should not be surprised if such a change does not alter the form of the equations of Lorentz, obviously independent of the choice of axes.

We are thus led to consider a continuous group which we call the Lorentz group and which admit as infinitesimal transformations:

1° the transformation T0 which is permutable with all others;

2° the three transformations T1, T2, T3;

3° the three rotations [T1, T2], [T2, T3], [T3, T1].

Any transformation of this group can always be decomposed into a transformation of the form:

But for our purposes, we should consider only a part of the transformations of this group; we must assume that l is a function of ε, and it is a question of choosing this function in such a way that this part of the group that I call P still forms a group.

Let's rotate the system 180° around the y-axis, we should find a transformation that will still belong to P. But this amounts to a sign change of x, x', z and z'; we find:

Consider the determinants that appear in both sides of (3) and at the begin of the first part; if we seek to develop, we see that the terms of the 2d and 3rd degree from ξ1, η1, ζ1 disappear and that the determinant is equal to

ω designates the radial component of the velocity ξ1, η1, ζ1, that is to say, the component directed along the radius vector indicating from point x, y, t to point x1, y1, z1.

In order to obtain the second determinant, I look at the coordinates of different molecules of the electron at instant t', which is the same for all molecules, but in such a way that for the molecule considered we have t1=t1′{\displaystyle t_{1}=t'_{1}}. The coordinates of a molecule will then be:

If we are dealing with a single electron, our integrals are reduced to a single element, provided we consider only the points x, y, x which are sufficiently remote so that r and ω have substantially the same value for all points of the electron. The potentials ψ, F, G, H depend on the position of the electron and also its velocity, because not only ξ1, η1, ζ1 show up in the numerator of F, G, H, but the radial component ω shows up in the denominator. It is of course its position and its velocity at the instant t1.

The partial derivatives of φ, F, G, H with respect to t, x, y, z (and therefore the electric and magnetic fields) will also depend on its acceleration. Moreover, they depend linearly, since the acceleration in these derivatives is introduced as a result of a single differentiation.

Langevin was thus led to distinguish the electric and magnetic field terms which do not depend on the acceleration (this is what he calls the velocity wave) and those that are proportional to acceleration (that is what he calls the acceleration wave).

The calculation of these two waves is facilitated by the Lorentz transformation. Indeed, we can apply this transformation to the system, so that the velocity of the single electron under consideration becomes zero. We will use for the x-axis the direction of the velocity before the transformation, so that, at the instant t1,

η1=ζ1=0{\displaystyle \eta _{1}=\zeta _{1}=0\,}

and we will take ε = -ξ, so that

ξ1′=η1′=ζ1′=0{\displaystyle \xi '_{1}=\eta '_{1}=\zeta '_{1}=0\,}

We can therefore reduce the computation of the two waves to the case where the electron velocity is zero. Let's start with the velocity wave, we first note that this wave is the same as if the electron motion was uniform.

If the electron continues to move in a rectilinear and uniform way with the velocity it had at the instant t1, that is to say, with the velocity -ε, 0, 0, the point (5) would be the one occupied at the instant t.

Taking the acceleration wave, we can, through the Lorentz transformation, reduce its determination to the case of zero velocity. This is the case if we imagine an electron whose oscillation amplitude is very small, but very fast, so that the displacements and velocities are much smaller, but the accelerations are finished. We thus come back to the field that has been studied in the famous work by Hertz entitled Die Kräfte elektrischer Schwingungen nach der Maxwell'schen Theorie, and that for a point at great distance. In these conditions:

I° Both electric and magnetic fields are equal.

2° They are perpendicular to each other.

3° They are perpendicular to the normal of the spherical wave, that is to say to the sphere whose center is the point x1, y1, z1.

I say that these three properties will remain, even when the velocity is not zero, and for this it is enough to show that they are not altered by the Lorentz transformation.

They then vanish in virtue of equations ∑f(x−x1)=∑α(x−x1)=0{\displaystyle \sum f\left(x-x_{1}\right)=\sum \alpha \left(x-x_{1}\right)=0} and in virtue of equations (6). Yet this is precisely what was demonstrated.

So the derivatives of ψ, F, G, H with respect to x, y, z, t (and hence also the two fields f, g, h; α, β, γ) will be homogeneous of degree -2 with respect to the same quantities, if we remember also that the relation

But these derivatives depend on these fields of x - x1, the velocities dx1dt1{\displaystyle {\frac {dx_{1}}{dt_{1}}}}, and the accelerations d2x1dt12{\displaystyle {\frac {d^{2}x_{1}}{dt_{1}^{2}}}}; they consist of a term independent of accelerations (velocity wave) and a term linear in respect to accelerations (acceleration waves). But dx1dt1{\displaystyle {\frac {dx_{1}}{dt_{1}}}} is homogeneous of degree 0 and d2x1dt12{\displaystyle {\frac {d^{2}x_{1}}{dt_{1}^{2}}}} is homogeneous of degree -1; hence it follows that the velocity wave is homogeneous of degree -2 with respect to x - x1, y - y1, z - z1, and the acceleration wave is homogeneous of degree -1. So in a very distant point an acceleration wave is predominant and can therefore be regarded as being assimilated to the total wave. In addition, the law of homogeneity shows that the acceleration wave is similar to itself at a distance and at any point. It is therefore, at any point, similar to the total wave at a remote point. But in a distant point the disturbance can propagate as plane waves, so that the two fields should be equal, mutually perpendicular and perpendicular to the direction of propagation.

I shall refer for more details to a work by Langevin in the Journal de Physique (Year 1905).

Suppose a single electron in rectilinear and uniform motion. From what we have seen, we can, through the Lorentz transformation, reduce the study of the field determined by the electron to the case where the electron is motionless; the Lorentz transformation replaces the real electron in motion by an ideal electron without motion.

Let α, β, γ, f, g, h be the real field; let α', β', γ', f', g', h' be the field after the Lorentz transformation, so the ideal field α', f' corresponds to the case where the electron is motionless; we have:

We now determine the total energy due to the motion of the electron, the corresponding action, and the electromagnetic momentum, in order to calculate the electromagnetic mass of the electron. For a distant point, it suffices to consider the electron as reduced to a single point; we are thus brought back to the formulas (4) of the preceding § which generally can be appropriate. But here they do not suffice, because the energy is mainly located in the ether parts nearest to the electron.

On this subject we can make several hypotheses.

According to that of Abraham, the electrons are spherical and not deformable.

So when we apply the Lorentz transformation when the real electron is spherical, the electron becomes a perfect ellipsoid. The equation of this ellipsoid is based on § 1:

If the radius of the real electron is r, the axes of the ideal electron would therefore be:

klr,lr,lr.{\displaystyle klr,\ lr,\ lr.}

In Lorentz's hypothesis, however, the moving electrons are deformed, so that the real electron would become a ellipsoid, while the ideal electron is still always a perfect sphere of radius r; the axes of the real electron will then be:

In Lorentz's hypothesis we have B' = 2A', and A ' (being inversely proportional to the radius of the electron) is a constant independent of the velocity of the real electron; we get for the total energy:

This granted, imagine that the electron is subject to a binding, so there is a relation between r and φ; in the hypothesis of Lorentz this relation would be φr = const., in that of Langevin φ²r² = const. We assume in a more general way

We must therefore have m=−23{\displaystyle m=-{\tfrac {2}{3}}} in conformity with the hypothesis of Langevin.

This result should come nearer to that which is connected to the first equation (a), and from which actually it does not differ. Indeed, suppose that every element dτ of the electron is subjected to a force Xdτ parallel to the x-axis, X is the same for all elements; we will then have, in conformity with the definition of momentum:

The conclusion is that if the electron is subject to a binding between its three axes, and if no other force intervenes except the binding forces, the shape of that electron, when it is given a uniform velocity, may be such that the ideal electron corresponds to a sphere, except the case where the binding is such that the volume is constant, in conformity with the hypothesis of Langevin.

We are led in this way to pose the following problem: what additional forces, other than the binding forces, are necessary to intervene to account for the law of Lorentz or, more generally, any law other than that of Langevin?

The simplest hypothesis, and the first that we should consider, is that these additional forces are derived from a special potential depending on the three axes of the ellipsoid, and therefore on θ and on r; let F(θ, r) be the potential; in which case the action will be expressed:

It remains to see if this hypothesis of the contraction of electrons reflects the inability to demonstrate absolute motion, and I will begin by studying the quasi-stationary motion of an isolated electron, or which is subject only to the action of other distant electrons.

It is known that what is called quasi-stationary motion is the motion where the velocity changes are slow enough so that the electric and magnetic energy due to motion of the electron differ little from what they would be in uniform motion; we know also that Abraham derived the transverse and longitudinal electromagnetic masses from the notion of quasi-stationary motion.

where we consider for the moment only the electric and magnetic fields due to the motion of an electron. In the preceding §, by considering the motion as uniform, we regarded H as dependent from the velocity ξ, η, ζ of the electrons' center of gravity (the three components in the preceding §, had as values -ε, 0, 0) and the parameters r and θ that define the shape of the electron.

But if the motion is more uniform, H depend not only on the values of ξ, η, ζ, r, θ at the instant in question, but on values of these quantities at other instants which may differ in quantities of the same order as the time by light to travel from one point to another of the electron; in other words, H depend not only on ξ, η, ζ, r, θ, but on their derivatives of all orders with respect to time.

Well, the motion is said to be quasi-stationary when the partial derivatives of H with respect to the successive derivatives of ξ, η, ζ, r, θ are negligible compared to the partial derivatives of H with respect to the quantities ξ, η, ζ, r, θ themselves.

In these equations, F has the same meaning as in the preceding §, X, Y, Z are the components of the force acting on the electron: this force is solely due to electric and magnetic fields produced by other electrons.

This is why Abraham gave dDdV{\displaystyle {\tfrac {dD}{dV}}} the name longitudinal mass and DV{\displaystyle {\tfrac {D}{V}}} the name transverse mass; recall that D=dHdV.{\displaystyle D={\tfrac {dH}{dV}}.}

∂H∂V{\displaystyle {\tfrac {\partial H}{\partial V}}} represent the derivative with respect to V, after r and θ were replaced by their values as functions of V from the first two equations (1); we will also have, after the substitution,

H=+A1−V2.{\displaystyle H=+A{\sqrt {1-V^{2}}}.}

We choose units so that the constant factor A is equal to 1, and I pose 1−V2=h{\displaystyle {\sqrt {1-V^{2}}}=h}, hence:

Let us return now to equations (11bis) of § 1; we can regard X1, Y1, Z1 as having the same meaning as in equations (5). On the other hand, we have l = 1 and ρ′ρ=kμ{\displaystyle {\frac {\rho ^{\prime }}{\rho }}=k\mu }; these equations then become:

This shows that the equations of quasi-stationary motion are not altered by the Lorentz transformation, but it still does not prove that the hypothesis of Lorentz is the only one that leads to this result.

To establish this point, we will restrict ourselves, as Lorentz did, to certain particular cases; it will be obviously sufficient for us to show a negative proposal.

How do we first extend the hypotheses underlying the above calculation?

1° Instead of assuming l = 1 in the Lorentz transformation, we assume any l.

2° Instead of assuming that F is proportional to the volume, and hence that H is proportional to h, we assume that F is any function of θ and r, so that [after replacing θ and r with their values as functions of V, from the first two equations (1)] H is any function of V.

I note first that, assuming H = h, we must have l = 1; and in fact the equations (6) and (7) remain, except that the right-hand sides will be multiplied by 1l{\displaystyle {\tfrac {1}{l}}}; so do equations (9), except that the right-hand sides will be multiplied by 1l2{\displaystyle {\tfrac {1}{l^{2}}}}; and finally the equations (10), except that the right-hand sides will be multiplied by 1l{\displaystyle {\tfrac {1}{l}}}. If we want that the equations of motion are not altered by the Lorentz transformation that is to say that the equations (10) only differ from equations (5) by the accentuation of the letters, it must be assumed:

l = 1.

Suppose now that we have η = ζ = 0, where ξ = V, dξdt=dVdt{\displaystyle {\frac {d\xi }{dt}}={\frac {dV}{dt}}}; the equations (5) take the form:

As l should depend only on ε (since, if there are more electrons, l must be the same for all electrons whose velocities ξ may be different), this identity can take place only if we have:

m=1,l=1.{\displaystyle m=1,\ l=1.}

Thus Lorentz's hypothesis is the only one consistent with the inability to demonstrate absolute motion; if we accept this impossibility, we must admit that the moving electrons contract and become ellipsoids of revolution where two axes remain constant; it must be admitted, as we have shown in the previous §, the existence of an additional potential which is proportional to the volume of the electron.

The analysis of Lorentz is therefore fully confirmed, but we can better give us an account of the true reason of the fact which occupies us; and this reason must be sought in the considerations of § 4. The transformations that do not alter the equations of motion must form a group, and this can take place only if l = 1. As we do not recognize if an electron is at rest or in absolute motion, it is necessary that, when in motion, it undergoes a distortion that must be precisely that which imposes the corresponding transformation of the group.

it is convenient to add a term representing the additional potential F of § 6; this term will obviously have the form:

J1=∫∑(F)dt{\displaystyle J_{1}=\int \sum (F)dt}

where Σ(F) represents the sum of the additional potential due to the different electrons, each of which is proportional to the volume of the corresponding electron.

I write (F) in brackets to avoid confusion with the vector F, G, H.

The total action is then J + J1. We saw in § 3 that J is not altered by the Lorentz transformation, we must show now that it is the same for J1.

We have for one electron,

(F)=ω0τ{\displaystyle (F)=\omega _{0}\tau \,}

ω0 being a special coefficient of the electron and τ its volume; so I can write:

∑(F)=∫ω0dτ,{\displaystyle \sum (F)=\int \omega _{0}d\tau ,}

the integral has to be extended to the entire space, but so that the coefficient ω0 is zero outside the electrons, and that within each electron it is equal to the special coefficient of that electron. Then we have:

Now we have ω0 = ω'0; for if a point belong to an electron, the corresponding point after the Lorentz transformation still belongs to the same electron. On the other hand, we found in § 3;

dτ′dt′=l4dτdt{\displaystyle d\tau 'dt'=l^{4}d\tau \ dt}

and since we now assume l = 1

dτ′dt′=dτdt{\displaystyle d\tau 'dt'=d\tau \ dt}

We have therefore

J1=J1′{\displaystyle J_{1}=J'_{1}\,}.﻿ C.Q.F.D.

The theorem is thus general, it gives us at the same time a solution of the question we posed at the end of § 1: finding the complementary forces which are unaltered by the Lorentz transformation. The additional potential (F) satisfies this condition.

So we can generalize the result announced at the end of § 1 and write:

If the inertia of electrons is exclusively of electromagnetic origin, if they are only subject to forces of electromagnetic origin, or to forces generated by the additional potential (F), no experiment can demonstrate absolute motion.

So what are these forces that create the potential (F)? They can obviously be compared to a pressure which would reign inside the electron; all occurs as if each electron were a hollow capacity subjected to a constant internal pressure (volume independent); the work of this pressure would be obviously proportional to the volume changes.

In any case, I must observe that this pressure is negative. Remember the equation (10) of § 6, according to Lorentz's hypothesis we write:

F=Ar3θ2{\displaystyle F=Ar^{3}\theta ^{2}};

equations (11) of § 6 give us:

A=a3b4.{\displaystyle A={\frac {a}{3b^{4}}}.}

Our pressure is equal to A, with a constant coefficient, which is indeed negative.

Now assessing the mass of the electron – I mean the "experimental mass", that is to say the mass for low velocities – we have (cf. § 6):

so that the mass, both longitudinal and transverse, will be ab{\displaystyle {\tfrac {a}{b}}}.

Now a is a numerical constant which shows that: the pressure that creates our additional potential is proportional to the 4th power of the experimental mass of the electron.

As Newton's law is proportional to the experimental mass, we are tempted to conclude that there is some relation between the cause that generates gravitation and the one that generates the additional potential.

Thus Lorentz's theory would completely explain the impossibility to demonstrate absolute motion, if all forces are of electromagnetic origin.

But there are forces which we can not assign an electromagnetic origin, as for example gravitation. It could happen, indeed, that two systems of bodies produce equivalent electromagnetic fields, that is to say, exerting the same action on the electrified bodies and on the currents, and yet these two systems do not exercise the same gravitational action on the Newtonian mass. The gravitational field is thus distinct from the electromagnetic field. Lorentz was thus forced to complete his hypothesis by assuming that forces of any origin, and in particular gravitation, are affected by translation (or, if preferred, by the Lorentz transformation) the same way as electromagnetic forces.

It is now convenient to enter into details and look more closely at this hypothesis. If we want that the Newtonian force is affected in this way by the Lorentz transformation, we can not accept that the force depends only on the relative position of the attracting body and of the body attracted at the instant considered. It will also depend on the velocities of the two bodies. And that's not all: it is natural to assume that the force acting at time t on the attracted body, depends on the position and velocity of this body at the same time t; but it will depend, in addition, on the position and velocity of the attracting body, not at time t, but a moment earlier, as if gravitation needs a certain time to propagate.

Consider therefore the position of the attracted body at the instant t0 and, at this point, x0, y0, z0 are the coordinates, ξ, η, ζ the components of its velocity; consider the other attracting body at the corresponding time t0 + t and, at this point, x0 + x, y0 + y, z0 + z are the coordinates, ξ1, η1, ζ1 the components of its velocity.

to define the time t. This relation will define the law of propagation of the gravitational action (I do not impose on me the condition that the propagation takes place with the same speed in all directions).

Now let X1, Y1, Z1 the 3 components of the action exerted at time t0 on the body; we have to express X1, Y1, Z1 as functions of

1° The condition (1) shall not be altered by transformations of the Lorentz group.

2° The components X1, Y1, Z1 will be affected by the Lorentz transformations the same way as electromagnetic forces designated by the same letters, that is to say, according to equations (11bis) of § 1.

3° When two bodies are at rest, we will fall back to the ordinary law of attraction.

It is important to note that in the latter case, the relation (1) disappears, because time does not play any role if the two bodies are at rest.

The problem thus posed is obviously undetermined. We will thus seek to satisfy as many as possible other additional conditions:

4° Astronomical observations do not appear to show significant derogation to Newton's law, we will choose the solution that deviates the least of this law, for low velocities of two bodies.

5° We will endeavor to arrange that T is always negative; if indeed it is conceived that the effect of gravitation takes a certain time to be propagated, it would be more difficult to understand how this effect could depend on the position not yet attained by the attracting body.

There is one case where the indeterminacy of the problem disappears; it is where the two bodies are at rest relative to each other, that is to say that:

It seems at first sight that the indetermination remains, since we have made no hypothesis about the value of t, that is to say about the speed of transmission; and that also x is a function of t, but it is easy to see that x - ξt, y, z (which appear in our formulas) do not depend on t.

We see that if two bodies are simply in motion by a common translation, the force acting on the body is drawn normal to an ellipsoid with its center at the attracting body.

To go further we must look for the invariants of the Lorentz group.

We know that the substitutions of this group (assuming l = 1) are linear substitutions which do not affect the quadratic form

as the coordinates of three points P, P', P" in a 4-dimensional space. We see that the Lorentz transformation is a rotation of that space around the origin, regarded as fixed. We shall therefore have no other distinct invariants than 6 distances of the 3 points P, P', P" between them and the origin, or, if you like it better, than the 2 expressions:

or the 4 expressions of the same form, deduced from permuting (in an arbitrary way) the three points P, P', P".

But what we look for are the functions of 10 variables (2) that are invariants; so we must, among the combinations of our 6 invariants, seek those which depend only on these 10 variables, that is to say those that are homogeneous of degree 0 as compared to δx, δy, δz, δt, as compared to δ1x, δ1y, δ1z, δ1t. We will thus have 4 distinct invariants, which are:

Let us now consider the transformations undergone by the components of the force; resume the equations (11) of § 1, which relate not to the force X1, Y1, Z1, which we consider here, but to the force X, Y, Z referred to unit volume. We pose also:

1° The left-hand side of relation (1), which defines the velocity of propagation must be a function of the four invariants (5)

One can obviously make a lot of hypotheses, we only look at two:

A) It may be

∑x2−t2=r2−t2=0,{\displaystyle \sum x^{2}-t^{2}=r^{2}-t^{2}=0,}

where t = ±r, and since t must be negative, t = -r. This means that the propagation velocity is equal to that of light. At first it seems that this hypothesis should be rejected without consideration. Laplace has indeed shown that this propagation is either instantaneous, or much faster than light. But Laplace had considered the hypothesis of finite speed of propagation, ceteris non mutatis; here, however, this hypothesis is complicated by many others, and it may happen that there is a more or less perfect compensation, as the applications of the Lorentz transformation gave us already so many examples.

We must choose between these combinations, and secondly, in order to define X1, Y1, Z1 we need a third equation. For such a choice, we must endeavor to bring us closer as much as possible to the law to Newton. Let's see what happens when (always making t = -r ) we neglect the squares of the velocities ξ η etc.. The 4 invariants (5) then become:

But to be able to compare it with the law of Newton, another transformation is needed; here x0 + x, y0 + y, z0 + z are the coordinates of the attracting body at the instant t0 + x, and r=∑x2{\displaystyle r={\sqrt {\sum x^{2}}}}; in the law of Newton it is necessary to consider the coordinates x0 + x1, y0 + y1, z0 + z1 of the attracting body at the instant t0, and the distance r1=∑x12{\displaystyle r_{1}={\sqrt {\sum x_{1}^{2}}}}.

We can neglect the square of time t required for the propagation and therefore proceed as if the movement was uniform, then we have:

This solution is not unique. Indeed, let C be the fourth invariant (5), C - 1 is of the order of the square of ξ, and it is equal to (A - B)².

So we could add to the 2ds members of each of equations (8) a term consisting of C - 1 multiplied by an arbitrary function of A, B, C, and a term of the form of (A - B)² also multiplied by a function of A, B, C.

At first sight, the solution (8) seems the most straightforward, it may nevertheless be adopted and in effect – since M, N, P are functions of X1, Y1, Z1 and T1=∑X1ξ{\displaystyle T_{1}=\sum X_{1}\xi } – we can draw from these three equations (8) the values of X1, Y1, Z1, but in some cases these values become imaginary.

It is clear that if α, β, γ are invariants, X1, Y1, Z1, T1 satisfy the basic condition, that is to say, it will undergo, by the effect of the Lorentz transformations, a suitable linear substitution.

But for the equations (9) to be consistent, we must have:

∑X1ξ−T1=0,{\displaystyle \sum X_{1}\xi -T_{1}=0,}

which, by replacing X1, Y1, Z1, T1 by their values (9) and multiplying by k0², becomes:

(10)

−Aα−β−Cγ=0.{\displaystyle -A\alpha -\beta -C\gamma =0.\,}

What we want is, if we neglect the square of speed of light, the squares of the velocities ξ, etc., as well as the product of accelerations by the distances as we did above, so that the values of X1, Y1, Z1 remain in conformity with the law of Newton.

We must therefore choose, for the invariant α, one that reduces to −1r13{\displaystyle -{\tfrac {1}{r_{1}^{3}}}} to the order of approximation adopted, that is to say 1B3{\displaystyle {\tfrac {1}{B^{3}}}}